MATHEMATICS

FORM 4

Quadratic Expressions

&

Equations

2.1 QUADRATIC EXPRESSIONS (QE)

# Is an expression in the form of ax2

+ bx + c,

where a, b and c are constants, a ≠ 0 and x is

an unknown.

# Has : only one unknown

: the highest power of the unknown is 2

Eg : 3x2

+ 2x - 1

m2

+ 2m

5 - y2

9p2

2.2 FACTORISATION OF QUADRATIC EXPRESSIONS

Factorisation of a QE is the process of writing the expresion as a product of two linear expressions.

Four types of factorise QE of the form :

1.1) ax2

+ c

i) take out the HCF of a and c.

ii) write “what remains” as the second factor.

eg: a) 6x2

+ 8 = 2 ( 3x2

+ 4)

b) 9 + 6m2

= 3 (3 + 2m2

)

(HCF

)

1.2) ax2

+ bx

i) take out the unknown (x) and HCF of a and

b.

ii) write “what remains” as the second factor.

eg: a) 12x2

+ 9x = 3x ( 4x + 3)

b) 7y + 14y2

= 7y (1 + 2y )

unknown y

(HCF

)

2) px2

− q, where p and q are perfect square

i) write p = a2

and q = b2

.

ii) write the answer as (ax + b)(ax − b).

eg: a) x2

− 9 = x2

− 32

= (x + 3)(x−3)

b) 49 + 16y2

= 72

− 42

y2

= 72

− (4y)2

= (7 + 4y)(7 − 4y)

3) ax2

+ bx + c, where a, b and c are not equal to zero.

i) list down the pair of numbers p and q such that pq = c .

ii) select the pair of p and q such that p + q = b

iii) write the answer as (x + p)(x + q).

eg: a) x2

+ 7x + 10

= (x + 2)(x + 5)

pq 10

p 1 2

q 10 5

p+ q 11 7

Since b and c are positive the

values of p and q must also

positive

eg: b) x2

− 8x + 12

= (x − 2)(x − 6)pq 12

p −1 −2

q −12 −6

p+ q −13 −8

Since c is positive and b is

negative, that is, pq is positive

and p + q is negative,then the

values of p and q must be

negative.

eg: c) x2

+ 5x − 6

= (x − 1)(x + 6)

pq − 6

p 1 −1

q −6 6

p+ q −5 5

Since c is negative, that is, pq

is negative, then the values of

p and q must be of different

signs, where one is positive

and the other is negative.

eg: d) x2

− 3x − 18

= (x + 3)(x − 6)

e) 16 − 8x + x2

= x2

− 8x + 16

= (x − 4)(x − 4)

or = (x − 4)2

pq − 18

p 1 −1 2 −2 3

q −18 18 −9 9 −6

p+ q −17 17 −7 7 −3

pq 16

p −1 −2 −4

q −16 −8 −4

p+

q

−17 −10 −8

Other method that can be used to factorise

ax2

+ bx + c is called cross method.

The steps to be followed are:

i) Factorise a as m n, then factorise c as p q such that mq + np = b.

mx +p +npx (nx)(p)

nx +q +mqx (mx)(q)

mnx2

+pq +(mq + np)x

ii) Write the answer as (mx + p)(nx + q)

eg: 2x2

− 13x + 15

= (x − 3)(2x − 5)

eg: 8 + 2x − x2

= −x2

+ 2x + 8

= (−x + 4)(x + 2)

x − 5 −10x

2x −3 −3x

2x2

15 −13x

−x 2 4x

x 4 −2x

−x2

8 +2x

4) Factorise QE containing coefficients with

common factors.

To factorise completely a QE containing

coefficients with a common factor, take out

the HCF first before finding the other two

factors.

eg: 2x2

+ 16x + 24

= 2(x2

+ 8x + 12)

= 2(x + 2)(x + 6)

2.3 QUADRATIC EQUATIONS

A quadratic equation (QEq) with one unknown

has an equal sign and the highest power of the

unknown is 2.

eg: x2

+ 4x + 3 = 0

m2

= 4m

General form : ax2

+ bx + c = 0

eg: Write 4/x = 5 − x in general form.

x2

− 5x + 4 = 0

2.4 ROOTS OF QUADRATIC EQUATIONS

The value of the unknown which satisfy a QEq are

called the roots of the QEq.

Verifying roots :

- Substituting a given value for the unknown in QEq

to determine whether it is a root.

eg: 2y2

= 4 − 7y; y = 4, y = 1/2

Value of y Left hand side

(LHS)

Right hand side

(RHS)

conclusion

4 2(42

)= 32 4 − 7(4) = −24 LHS ≠ RHS

4 is not the root.

1/2 2(1/2 )2

= 1/2 4 − 7(1/2) = 1/2 LHS = RHS

1/2 is not the

root.

Determining roots by factorisation.

Steps :

1) express the equation in general form :

ax2

+ bx + c = 0

2) Factorise ax2

+ bx + c to express the equation in factor form (mx +p)(nx

+ q) = 0.

3) Write mx + p = 0 and nx + q = 0

4) Solve for x, x = − p/m and x = −q/n

Eg: w(w + 3) = 9(w − 1)

w2

+ 3w = 9w – 9

Step 1: w2

– 6w − 9 = 0

Step 2: (w – 3)(w − 3) = 0

Step 3: w − 3 = 0 and w − 3 = 0

Step 4: w = 3 [repeated root]

Eg: 3p2

+ 4p = 2

p + 4

3p2

+ 4p = 2(p + 4)

Step 1: 3p2

+ 2p − 8 = 0

Step 2: (3p – 4)(p + 2) = 0

Step 3: 3p − 4 = 0 and p + 2 = 0

Step 4: p = 4/3, p = − 2